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An equation is a mathematical sentence that contains an = sign. 555 = A-75 A-75 = 555 are examples of an equation. In the examples above “A” is a variable which represents an unknown number. One way to solve for A is to complete the Inverse (opposite) Operation (the opposite of subtraction is addition) 555 + 75 = A or A = 555 + 75, so A = 630 1. T - 5 = 11 11 + 5 = T 16 = 16 2. 16 - T = 11 Find the example that does not use the inverse Operation. T = 16 - 11 T=5 3. B + 6 = 15 4. 9 + B = 15 B = 15 - 6 B = 15 - 9 B=9 B=6 Memorize this type of problem so you can solve using the same operation 5. 6 x R = 48 48 6 = R 8=R 6. W x 8 = 48 48 8 = w 6=w Find the example that does not use the inverse Operation. 7. 72 S = 9 Memorize this type of 72 9 = S problem so 8=S you can solve using the same operation 8. M 9 = 8 8x9=M 72 = M Equations are like a Balance scale. If weight is added or subtracted to one side it will make the scale tip or not be balanced. In order to keep the scale balanced you must add or subtract equal amounts to both sides. This idea will help us solve equations with variables. Property of Equality for Addition and Subtraction F - 244 =120 F - 244 + 244 = 120 + 244 Add 244 to both sides Solve Addition Property of Equality If you add the same number to each side of an equation, then the 2 sides remain the same. F = 364 X + 3.6 = 12.4 Subtract 3.6 from both sides X + 3.6 - 3.6 = 12.4 - 3.6 Solve X = 8.8 Subtraction Property of Equality If you subtract the same number from each side of an equation, then the 2 sides remain the same. Property of Equality for Multiplication and Division 434 = 2s 434 = 2s 2 2 Remember a variable next to a number indicates multiplication S = 217 Divide both sides by 2 Solve Division Property of Equality If each side of an equation is divided by the same nonzero number, then the 2 sides remain equal. n 8 = 96 Multiply both sides by 8 n 8 x 8 = 96 x 8 Solve n = 768 Multiplication Property of Equality If each side of an equation is multiplied by the same number, then the 2 sides remain equal. An expression is one part of an equation Examples 15 - 6 20 + a Numerical expressions are often written in sentence form. 7-x 42 d Five more hits than the Yankees Ten fewer points than the Knicks Yankees + 5 Knicks - 10 or Y+5 K - 10 The key words often indicate what to do more means add fewer means subtract Addition Multiplication Plus Times Sum Product Subtraction multiplied More than Minus Increased by Difference Division Total Less than Divided Subtract quotient Decreased by 3x = 18 3x + 9 = 18 How are these equations different? When one side of an equation has 2 or more operations we need more than one step to solve. 3x + 9 = 18 First subtract 9 from both sides. 3x + 9 - 9 = 18 - 9 Simplify 3x = 9 Then divide each side by 3 3x = 9 3 3 So x = 3 B - 0.8 = 1.3 9 First add 0.8 to both sides B - 0.8 + 0.8 = 1.3 + 0.8 9 Simplify B = 2.1 9 B x 9 = 2.1 x 9 9 Then multiply both sides by 9 So B = 18.9 An integer is a whole number that can be either greater than 0, called positive, or less than 0, called negative. Zero is neither positive nor negative. Two integers that are the same distance from zero in opposite directions are called opposites. -5 is the opposite of 5 Every integer on the number line has an absolute value, which is its distance from zero. The brackets indicate the absolute value. Using a Number Line to Add or Subtract Integers Add a positive integer by moving to the right on the number line Add a negative integer by moving to the left on the number line 2+6=? 8 + (- 3 ) = ? 2 6 8 -3 3-7=? Subtract an integer by adding its opposite -7 3 3 - (-7) = ? 7 3 Another way to remember the rule for subtraction is Keep, change, change! 3- 7=? Keep the first number Change the sign + 3 Change the sign of the second number -7 4-9=? Keep the 4 4 + (-9) Change the sign Change the sign of the 9 To multiply or divide signed integers, always multiply or divide the absolute values and use these rules to determine the sign of the answer: If the signs are the same the product or quotient is Positive 6 x 3 = 18 (-6 ) x (-3) = 18 If the signs are different the product or quotient is Negative (-6 ) x 3 = -18 6 x (-3) = -18 Using Counters to Solve Integer Problems A positive and a negative counter equal 0 + _ To add we put counters in the box To add 4 + (-3) we start with 4 positive counters + + _ + _ + We put 3 negative counters in _ The positive and negative pairs cancel each other out to make 0 We are left with 1 positive counter Subtracting Integers With Counters To subtract we take counters out of the box 4 - (-5) + + _ + + _ + + _ + _ + + _ Since there are no negative counters we must add negatives to the box. If we add 0 to the box we do not change the value If we take out the 5 negatives we will be left with 9 positives + + + + + + + + + 4 - (-5) = 9 + _ Using Counters to Multiply Integers In the sentences 3 x 2 we will place 3 groups of 2 in the box + + + + + + + + + + + + Multiplying a Positive Integer by a Negative Integer In the problem 3 x (-2) we are putting in 3 groups of 2 negatives in the box _ _ _ _ _ _ _ _ _ _ _ _ The box shows that 3 x (-2) = - 6 Multiplying With a Negative Integer When multiplying by a negative integer we are taking out groups In the problem (-2 ) x 3 we are taking out 2 groups of 3 Since the box is empty we must add 0 pairs until we have enough to take out 2 groups of 3 + _ + _ + _ + _ + _ + _ + _ + _ + _ + _ + _ + _ We can now take out 2 groups of 3 (-2) x 3 = -6 _ _ _ _ _ _ We are left with 6 negatives 3x + 9 = 18 3x + 9 > 18 How are these number sentences different? When an equation has > or < sin it is called an inequality We solve inequalities in the same way as equations 3x + 9 > 18 First subtract 9 from both sides. 3x + 9 - 9 > 18 - 9 Simplify 3x > 9 Then divide each side by 3 3x > 9 3 3 So x > 3 We can graph the solution to an inequality on a number line. 2a - 5 > 9 First, solve as you would an equation 2a - 5 + 5 > 9 + 5 2a > 9 2 2 a > 2.5 Add 5 to both sides Divide both sides by 2 Since a is greater than 2.5 it will include all numbers greater than 2.5 but not 2.5 We can draw a circle at 2.5 to show this. coordinate plane The plane determined by a horizontal number line, called the x-axis,.. and a vertical number line, called the yaxis, Each point in the coordinate plane can be specified by an ordered pair of numbers (-3,1) intersecting at a point called the origin Here's one way geometry is used in the real world. A team of archaeologists is studying the ruins of Lignite, a small mining town from the 1800's. They plot points on a coordinate plane to show exactly where each artifact is found. (1,3) Name each Point. (-5,2) (-2,4) (3,-4) (5,-6) (5,-4)